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Enumerative tropical algebraic geometry in $\mathbb{R} ^2$

Author: Grigory Mikhalkin
Journal: J. Amer. Math. Soc. 18 (2005), 313-377
MSC (2000): Primary 14N35, 52B20; Secondary 14N10, 14P25, 51M20
Published electronically: January 20, 2005
MathSciNet review: 2137980
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Abstract: The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629-634.

The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the real space $\mathbb{R} ^n$ and holomorphic curves with certain piecewise-linear graphs there.

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Additional Information

Grigory Mikhalkin
Affiliation: Department of Mathematics, University of Toronto, 100 St. George St., Toronto, Ontario, M5S 3G3 Canada and St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011 Russia
Address at time of publication: IHES, Le Bois-Marie, 35, route de Chartres, Bures-sur-Yvette, 91440, France

Keywords: Tropical curves, enumerative geometry, Gromov-Witten invariants, toric surfaces
Received by editor(s): January 31, 2004
Published electronically: January 20, 2005
Additional Notes: The author would like to acknowledge partial support of the NSF and NSERC
Article copyright: © Copyright 2005 Grigory Mikhalkin

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